Edit detail for SandBox Quaternion Algebra is Frobenius in Many Ways revision 4 of 32

changed:
-4-dimensional vector space representing Quaternion algebra
-
Quaternion Algebra Is Frobenius In Many Ways

Linear operators over a 4-dimensional vector space representing quaternion algebra

Ref:

- http://arxiv.org/abs/1103.5113

  $S_3$-permuted Frobenius Algebras

  *Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)*

- http://mat.uab.es/~kock/TQFT.html

  Frobenius algebras and 2D topological quantum field theories

  *Joachim Kock*

- http://en.wikipedia.org/wiki/Frobenius_algebra

We need the Axiom LinearOperator library.
\begin{axiom}
)library MONAL PROP LIN CALEY
\end{axiom}

Use the following macros for convenient notation
\begin{axiom}
-- summation
macro Σ(x)==reduce(+,x)
macro ΣΞ(x,i)==reduce(+,[x for i in 1..dim])
-- list
macro Ξ(f,i)==[f for i in 1..dim]
\end{axiom}

𝐋 is the domain of 4-dimensional linear operators over the domain of rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

changed:
-T := CartesianTensor(1,dim,EXPR INT)
-X:List T := [unravel [(i=j => 1;0) for j in 1..dim] for i in 1..dim]
-X(1),X(2)
-\end{axiom}
-
-Generate structure constants for Quaternion Algebra
-\begin{axiom}
-B:=map(x+->quatern(x.1,x.2,x.3,x.4),1$SQMATRIX(4,FRAC INT)::List List FRAC INT)
-M:=matrix [[B.i*B.j for j in 1..4] for i in 1..4]                              
-S(y)==map(x+->(x*inv(y)=1 or x*inv(y)=-1 => x*inv(y);0),M)                     
-Yg:T:=unravel concat concat(map(S,B)::List List List FRAC POLY INT)
-\end{axiom}
macro ℒ == List
macro ℂ == CaleyDickson
macro ℚ == Expression Integer
𝐋 := LinearOperator(dim, OVAR [], ℚ)
𝐞:ℒ 𝐋      := basisVectors()
𝐝:ℒ 𝐋      := basisForms()
o:𝐋:=1     -- identity for product
I:𝐋:=[1]   -- identity for composition
X:𝐋:=[2,1] -- twist
\end{axiom}

Now generate structure constants for Quaternion Algebra

The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.
\begin{axiom}
-- Also split-complex via Caley-Dickson parameter (p0 = -1)
q0:=subscript('q,[0])
q1:=subscript('q,[1])
QQ := ℂ(ℂ(ℚ,'i,q0),'j,q1)
-- Basis: Each B.i is a quaternion number
B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ)
-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j,i),j)
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real(x/y),M)
-- The result is a nested list
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
-- structure constants form a tensor operator
Y := ΣΞ(ΣΞ(ΣΞ(ѕ(i)(j)(k)*𝐞.i*𝐝.j*𝐝.k,i),j),k)
arity Y
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y,i),j)
\end{axiom}

changed:
-U:T := unravel(concat
-  [[script(u,[[],[j,i]])
-    for i in 1..dim]
-      for j in 1..dim]
-        )
-\end{axiom}
U:=inp([inp([script(u,[[j,i],[]]) for i in 1..dim])$𝐋 for j in 1..dim])$𝐋
\end{axiom}


changed:
-  In other words, if the (3,0)-tensor::
-
-    i  j  k   i  j  k   i  j  k
-     \ | /     \/  /     \  \/
-      \|/   =   \ /   -   \ /
-       0         0         0
  In other words, if the (3,0)-tensor:
$$
\scalebox{1} % Change this value to rescale the drawing.
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\usefont{T1}{ptm}{m}{n}
\rput(1.6948438,0.205){=}
\end{pspicture} 
}
$$

changed:
-\begin{axiom}
-ω := reindex(reindex(U,[2,1])*reindex(Yg,[1,3,2]),[3,2,1])-U*Yg
-\end{axiom}
Using the LinearOperator domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.

\begin{axiom}

ω:𝐋 :=
     o    Y I      /
     o     U       -
     o    I Y      /
     o     U       o

\end{axiom}

Note: The only purpose of the o symbols on the left above is to serve as a constant left-side margin as required by Axiom. The symbols on the right describe the relation between row.


changed:
-  is called *pre-Frobenius*.
  is called a [Frobenius Algebra].

added:


changed:
-J := jacobian(ravel ω,concat(map(variables,ravel U))::List Symbol);
-uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];
-J::OutputForm * uu::OutputForm = 0
-nrows(J)
-ncols(J)
-\end{axiom}
J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);
u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];
J::OutputForm * u::OutputForm = 0
nrows(J),ncols(J)
\end{axiom}


changed:
-\begin{axiom}
-NJ:=nullSpace(J)
-SS:=map((x,y)+->x=y,concat map(variables,ravel U),
-  entries reduce(+,[p[i]*NJ.i for i in 1..#NJ]))
-Ug:T := unravel(map(x+->subst(x,SS),ravel U))
-\end{axiom}

\begin{axiom}
Ñ:=nullSpace(J)
ℰ:=map((x,y)+->x=y,concat map(variables,ravel U),
  entries Σ[p[i]*Ñ.i for i in 1..#Ñ])
Ų := map(x+->subst(x,ℰ),U)$𝐋
\end{axiom}

changed:
-test(unravel(map(x+->subst(x,SS),ravel ω))$T=0*ω)
-\end{axiom}
test(map(x+->subst(x,ℰ),ω)$𝐋=0*ω)
\end{axiom}

changed:
-Ud:DMP([p[i] for i in 1..#NJ],INT) := determinant [[Ug[i,j] for j in 1..dim] for i in 1..dim]
-factor Ud
-\end{axiom}
Ů:=determinant [[retract((𝐞.i * 𝐞.j)/Ų) for j in 1..dim] for i in 1..dim]
factor Ů
\end{axiom}

changed:
-\begin{axiom}
-Ωg:T:=unravel concat(transpose(1/Ud*adjoint([[Ug[i,j] for j in 1..dim] for i in 1..dim]).adjMat)::List List FRAC POLY INT)
-\end{axiom}
-<center><pre>
-dimension
-Ω
-U
-</pre></center>
-\begin{axiom}
-contract(contract(Ωg,1,Ug,1),1,2)
-\end{axiom}

\begin{axiom}
Ω:𝐋:=unravel((0/2)$Prop,concat(transpose(1/Ů*adjoint([[retract((𝐞.i * 𝐞.j)/Ų)
  for j in 1..dim] for i in 1..dim]).adjMat)::ℒ ℒ ℚ))
\end{axiom}

Check dimension
\begin{axiom}

d:𝐋:=
    o   Ω    /
    o   Ų    o

\end{axiom}

changed:
-  Co-multiplication
-\begin{axiom}
-λg:=reindex(contract(contract(Ug*Yg,1,Ωg,1),1,Ωg,1),[2,3,1]);
--- just for display
-reindex(λg,[3,1,2])
-\end{axiom}
-<center><pre>
-i  
-λ=Ω
-</pre></center>
-\begin{axiom}
-test(λg*X(1)=Ωg)
-\end{axiom}
  Co-algebra::

    λ:𝐋 :=
         o    Ω Ω  I    /
         o   I Y I I    /
         o   I  X  I    /
         o   I I  Ų     o
    --test

Why aren't these the same??

\begin{axiom}

λ:𝐋 :=
     o    I Ω     /
     o     Y I    o

λ2:𝐋 :=
     o     Ω I    /
     o    I Y     o

λ - λ2

\end{axiom}

i = Unit of the algebra
\begin{axiom}
i:=𝐞.1
test
     o    i     /
     o    λ     =    Ω

\end{axiom}

changed:
-ιg:=X(1)*Ug

ι:𝐋:=
    o    i I    /
    o     Ų    o


changed:
-test(ιg * Yg = Ug)
-\end{axiom}
test
   o     Y     /
   o     ι     o  = Ų

\end{axiom}

changed:
-Ug0:T:=unravel eval(ravel Ug,[p[1]=1,p[2]=0,p[3]=0,p[4]=1])
-Ωg0:T:=unravel eval(ravel Ωg,[p[1]=1,p[2]=0,p[3]=0,p[4]=1])
-λg0:T:=unravel eval(ravel λg,[p[1]=1,p[2]=0,p[3]=0,p[4]=1]);
-reindex(λg0,[3,1,2])
-\end{axiom}
-
-$S_3$-permuted Frobenius Algebras
-
-    Zbigniew Oziewicz, Gregory Peter Wene
-    (26 Mar 2011)
-    http://arxiv.org/abs/1103.5113
-
-\begin{axiom}
-test( Yg = reindex(reindex(  reindex(Ug*Yg,[1,2,3]),  [2,3,1])*Ωg,[3,1,2]) )
-
-Yg213 := reindex(reindex(  reindex(Ug*Yg,[2,1,3]),  [2,3,1])*Ωg,[3,1,2]);
-ω213  := reindex(reindex(U,[2,1])*reindex(Yg213,[1,3,2]),[3,2,1])-U*Yg213;
-J213  := jacobian(ravel ω213,concat(map(variables,ravel U))::List Symbol);
-NJ213 := nullSpace(J213)
-
--- opposite algebra
-Yg132 := reindex(reindex(  reindex(Ug*Yg,[1,3,2]),  [2,3,1])*Ωg,[3,1,2]);
-ω132  := reindex(reindex(U,[2,1])*reindex(Yg132,[1,3,2]),[3,2,1])-U*Yg132;
-J132  := jacobian(ravel ω132,concat(map(variables,ravel U))::List Symbol);
-NJ132 := nullSpace(J132)
-
-Yg321 := reindex(reindex(  reindex(Ug*Yg,[3,2,1]),  [2,3,1])*Ωg,[3,1,2]);
-ω321  := reindex(reindex(U,[2,1])*reindex(Yg321,[1,3,2]),[3,2,1])-U*Yg321;
-J321  := jacobian(ravel ω321,concat(map(variables,ravel U))::List Symbol);
-NJ321 := nullSpace(J321)
-
-Yg312 := reindex(reindex(  reindex(Ug*Yg,[3,1,2]),  [2,3,1])*Ωg,[3,1,2]);
-ω312  := reindex(reindex(U,[2,1])*reindex(Yg312,[1,3,2]),[3,2,1])-U*Yg312;
-J312  := jacobian(ravel ω312,concat(map(variables,ravel U))::List Symbol);
-NJ312 := nullSpace(J312)
-
-Yg231 := reindex(reindex(  reindex(Ug*Yg,[2,3,1]),  [2,3,1])*Ωg,[3,1,2]);
-ω231  := reindex(reindex(U,[2,1])*reindex(Yg231,[1,3,2]),[3,2,1])-U*Yg231;
-J231  := jacobian(ravel ω231,concat(map(variables,ravel U))::List Symbol);
-NJ231 := nullSpace(J231)
-\end{axiom}
Ų0:=map(x+->subst(x,[q[0]=-1,q[1]=-1,p[1]=1,p[2]=1,p[3]=1,p[4]=-sqrt(2)]),Ų)$𝐋
Ω0:=map(x+->subst(x,[q[0]=-1,q[1]=-1,p[1]=1,p[2]=1,p[3]=1,p[4]=-sqrt(2)]),Ω)$𝐋
λ0:=map(x+->subst(x,[q[0]=-1,q[1]=-1,p[1]=1,p[2]=1,p[3]=1,p[4]=-sqrt(2)]),λ)$𝐋
\end{axiom}

Quaternion Algebra Is Frobenius In Many Ways

Linear operators over a 4-dimensional vector space representing quaternion algebra

Ref:

• http://arxiv.org/abs/1103.5113

  -permuted Frobenius Algebras

  Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)

• http://mat.uab.es/~kock/TQFT.html

  Frobenius algebras and 2D topological quantum field theories

  Joachim Kock

• http://en.wikipedia.org/wiki/Frobenius_algebra

We need the Axiom LinearOperator? library.

axiom

)library MONAL PROP LIN CALEY

   Monoidal is now explicitly exposed in frame initial 
   Monoidal will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/MONAL.NRLIB/MONAL
   Prop is now explicitly exposed in frame initial 
   Prop will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/PROP.NRLIB/PROP
   LinearOperator is now explicitly exposed in frame initial 
   LinearOperator will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/LIN.NRLIB/LIN
   CaleyDickson is now explicitly exposed in frame initial 
   CaleyDickson will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY

Use the following macros for convenient notation

axiom

-- summation
macro Σ(x)==reduce(+,x)

Type: Void

axiom

macro ΣΞ(x,i)==reduce(+,[x for i in 1..dim])

Type: Void

axiom

-- list
macro Ξ(f,i)==[f for i in 1..dim]

Type: Void

𝐋 is the domain of 4-dimensional linear operators over the domain of rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

axiom

dim:=4

(1)

Type: PositiveInteger?

axiom

macro ℒ == List

Type: Void

axiom

macro ℂ == CaleyDickson

Type: Void

axiom

macro ℚ == Expression Integer

Type: Void

axiom

𝐋 := LinearOperator(dim, OVAR [], ℚ)

(2)

Type: Type

axiom

𝐞:ℒ 𝐋      := basisVectors()

(3)

Type: List(LinearOperator?(4,OrderedVariableList?([]),Expression(Integer)))

axiom

𝐝:ℒ 𝐋      := basisForms()

(4)

Type: List(LinearOperator?(4,OrderedVariableList?([]),Expression(Integer)))

axiom

o:𝐋:=1     -- identity for product

(5)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

axiom

I:𝐋:=[1]   -- identity for composition

(6)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

axiom

X:𝐋:=[2,1] -- twist

(7)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

Now generate structure constants for Quaternion Algebra

The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

axiom

-- Also split-complex via Caley-Dickson parameter (p0 = -1)
q0:=subscript('q,[0])

(8)

Type: Symbol

axiom

q1:=subscript('q,[1])

(9)

Type: Symbol

axiom

QQ := ℂ(ℂ(ℚ,'i,q0),'j,q1)

?}\ } (\hbox{\axiomType{CaleyDickson?}\ } (\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , i , 01 q (0)) , j , 01 q (1))" class="equation" src="images/8228930438219976963-16.0px.png" align="bottom" Style="vertical-align:text-bottom" width="551" height="18"/>

(10)

Type: Type

axiom

-- Basis: Each B.i is a quaternion number
B:ℒ QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::ℒ ℒ ℚ)

(11)

Type: List(CaleyDickson?(CaleyDickson?(Expression(Integer),i,*01q(0)),j,*01q(1)))

axiom

-- Multiplication table:
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j,i),j)

(12)

Type: Matrix(CaleyDickson?(CaleyDickson?(Expression(Integer),i,*01q(0)),j,*01q(1)))

axiom

-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real(x/y),M)

Type: Void

axiom

-- The result is a nested list
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ

axiom

Compiling function S with type CaleyDickson(CaleyDickson(Expression(
      Integer),i,*01q(0)),j,*01q(1)) -> Matrix(Expression(Integer))

(13)

Type: List(List(List(Expression(Integer))))

axiom

-- structure constants form a tensor operator
Y := ΣΞ(ΣΞ(ΣΞ(ѕ(i)(j)(k)*𝐞.i*𝐝.j*𝐝.k,i),j),k)

(14)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

axiom

arity Y

(15)

Type: Prop

axiom

matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y,i),j)

(16)

Type: Matrix(LinearOperator?(4,OrderedVariableList?([]),Expression(Integer)))

A scalar product is denoted by the (2,0)-tensor

axiom

U:=inp([inp([script(u,[[j,i],[]]) for i in 1..dim])$𝐋 for j in 1..dim])$𝐋

(17)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

Definition 1

We say that the scalar product is associative if the tensor equation holds:

    Y   =   Y
     U     U

In other words, if the (3,0)-tensor:

(18)

(three-point function) is zero.

Using the LinearOperator? domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.

axiom

ω:𝐋 :=
     o    Y I      /
     o     U       -
     o    I Y      /
     o     U       o

(19)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

Note: The only purpose of the o symbols on the left above is to serve as a constant left-side margin as required by Axiom. The symbols on the right describe the relation between row.

Definition 2

An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.

We may consider the problem where multiplication Y is given, and look for all associative scalar products

This problem can be solved using linear algebra.

axiom

)expose MCALCFN

   MultiVariableCalculusFunctions is now explicitly exposed in frame 
      initial 
J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);

Type: Matrix(Expression(Integer))

axiom

u := transpose matrix [concat map(variables,ravel U)::ℒ Symbol];

Type: Matrix(Polynomial(Integer))

axiom

J::OutputForm * u::OutputForm = 0

(20)

Type: Equation(OutputForm?)

axiom

nrows(J),ncols(J)

(21)

Type: Tuple(PositiveInteger?)

The matrix J transforms the coefficients of the tensor into coefficients of the tensor . We are looking for the general linear family of tensors such that J transforms into for any such .

If the null space of the J matrix is not empty we can use the basis to find all non-trivial solutions for U:

axiom

Ñ:=nullSpace(J)

(22)

Type: List(Vector(Expression(Integer)))

axiom

ℰ:=map((x,y)+->x=y,concat map(variables,ravel U),
  entries Σ[p[i]*Ñ.i for i in 1..#Ñ])

(23)

Type: List(Equation(Expression(Integer)))

axiom

Ų := map(x+->subst(x,ℰ),U)$𝐋

(24)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

This defines a family of pre-Frobenius algebras:

axiom

test(map(x+->subst(x,ℰ),ω)$𝐋=0*ω)

(25)

Type: Boolean

The scalar product must be non-degenerate:

axiom

Ů:=determinant [[retract((𝐞.i * 𝐞.j)/Ų) for j in 1..dim] for i in 1..dim]

(26)

Type: Expression(Integer)

axiom

factor Ů

(27)

Type: Factored(Expression(Integer))

Definition 3

Co-pairing

axiom

Ω:𝐋:=unravel((0/2)$Prop,concat(transpose(1/Ů*adjoint([[retract((𝐞.i * 𝐞.j)/Ų)
  for j in 1..dim] for i in 1..dim]).adjMat)::ℒ ℒ ℚ))

(28)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

Check dimension

axiom

d:𝐋:=
    o   Ω    /
    o   Ų    o

(29)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

Definition 4

Co-algebra:

    λ:𝐋 :=
         o    Ω Ω  I    /
         o   I Y I I    /
         o   I  X  I    /
         o   I I  Ų     o
    --test

Why aren't these the same??

axiom

λ:𝐋 :=
     o    I Ω     /
     o     Y I    o

(30)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

axiom

λ2:𝐋 :=
     o     Ω I    /
     o    I Y     o

(31)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

axiom

λ - λ2

(32)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

i = Unit of the algebra

axiom

i:=𝐞.1

(33)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

axiom

test
     o    i     /
     o    λ     =    Ω

(34)

Type: Boolean

Definition 5

Co-unit

  i 
  U
  

axiom

ι:𝐋:=
    o    i I    /
    o     Ų    o

(35)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

Y=U
ι  

axiom

test
   o     Y     /
   o     ι     o  = Ų

(36)

Type: Boolean

For example:

axiom

Ų0:=map(x+->subst(x,[q[0]=-1,q[1]=-1,p[1]=1,p[2]=1,p[3]=1,p[4]=-sqrt(2)]),Ų)$𝐋

(37)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

axiom

Ω0:=map(x+->subst(x,[q[0]=-1,q[1]=-1,p[1]=1,p[2]=1,p[3]=1,p[4]=-sqrt(2)]),Ω)$𝐋

(38)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))

axiom

λ0:=map(x+->subst(x,[q[0]=-1,q[1]=-1,p[1]=1,p[2]=1,p[3]=1,p[4]=-sqrt(2)]),λ)$𝐋

(39)

Type: LinearOperator?(4,OrderedVariableList?([]),Expression(Integer))